I'm grading a complex analysis course right now and it turns out to involve a lot of contour integration. For instance, students are asked to find the integral
$$\int_0^\infty \frac{\cos (ax)}{(x^2 ...

Reading through Titchmarh's book on the Riemann Zeta Function, chapter 3 discusses the Prime Number Theorem. One way to prove this result is to check the zeta function has no zeros on the line $z = 1 ...

I am a physicist who usually doesn't need to care about the fact that square root is not single-valued on the complex plane. But I would like to give a meaning to and compute the contour integrals :
...

I have below function to integrate;
$$\int_{0}^{\infty} \frac{J_{0}(ax)x^3}{k^2-x^2} dx$$ here $a,k$ are constants.
Since this is an odd function, I am not allowed to extend the limits from negative ...

When solving an improper integration from $0$ to $\infty$ which involves an even function, the integration limits can be extended from $-\infty$ to $\infty$. For example consider even function $f(x)$; ...

I am trying to solve below integration;
$$\int_{0}^{\infty} H_{0}^{1}(pR)\sin(pR)\frac{p}{k^2-p^2} dp$$
here $k,R$ are constants. This is related to the question link.
Below shows my approach to get ...

I want to solve the integral attached below by means of residue theorem. I tried the common integration ways and seeked references(e.g, Rjadov, et. al).
Finally, I decided to solve this integral by ...

Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be two arbitrary functions. Assume $g\in L^2(\mathbb{R})$.
I'm looking to find out the minimal set of assumptions on $f$ and $g$ such ...

Given the series
\begin{align}
S_{x}(a) = \sum_{k=1}^{\infty} (-1)^{k+1} \, \binom{x-1}{k} \, L_{k+n-1}(a)
\end{align}
where $L_{m}(x)$ is the Laguerre polynomial.
By using
\begin{align}
L_{n}(z) = ...

I have to solve the next integral:
$$\int_{-\infty}^{\infty} e^{ibx}(e^{ia/x}-1)dx$$ where $a,b$ are real parameters.
I can use Jordan´s Theorem to show that as $f(z)=e^{ibz}g(z)$ where $g(z)=(e^{ ia ...

Let $D$ be the annulus $6\lt |z| \lt 8$ and let $C$ be any simple closed contour inside $D$. Show that there holds:
$$\int_C \frac{dz}{z^2+1} =0$$
This has two singular points, $z=\pm i$, these are ...

Suppose we want to find the asymptotic behavior as $n \rightarrow \infty$ of the integral
$$\int_C \frac{dz}{z} \frac{e^z}{z^n}=\int_C \frac{dz}{z} \exp(z-n \ln z)$$
where $C$ is some contour in the ...

I want to evaluate the following:
$$\int_C \frac{\sin z}{(z+1)^7} \mathrm{d}z$$
Where $C$ is the circle of radius $5$, centre $0$, positively oriented.
Now this has one root at $z=-1$. Now I should ...